Twist moves and the affine index polynomials of virtual knots

Abstract

In this paper, we give a necessary condition for two virtual knots to be related by a finite sequence of twist moves by using the affine index polynomial, which is a Vassiliev invariant of degree [Formula: see text]. Trapp showed that a numerical Vassiliev invariant of degree [Formula: see text] has a polynomial growth of degree [Formula: see text] on a twist sequence of knots, which can be extended to a twist sequence of virtual knots. We calculate the growth of the affine index polynomial for a twist sequence of virtual knots and find the difference of the affine index polynomials of two virtual knots, which are related by a twist move. Moreover, we give a lower bound for the number of twist moves needed to transform [Formula: see text] to [Formula: see text] if [Formula: see text] and [Formula: see text] are virtual knots related by a finite sequence of twist moves.